Point degree spectra of represented spaces Arno Pauly Swansea - - PowerPoint PPT Presentation

Point degree spectra of represented spaces

Arno Pauly

Swansea University

Joint work with Takayuki Kihara, Nagoya

The talk in a nutshell

The notion of a point degree spectrum of a space links: ◮ Descriptive Set Theory ◮ Computable Analysis ◮ Recursion Theory and allows us to use techniques from one field to solve problems in others.

Small inductive dimension

Definition

Let the (small inductive) dimension of a Polish space be defined inductively via dim(∅) = −1 and: dim(X) = sup

x∈X

sup

n∈N

inf

U∈O(X),x∈U⊆B(x,2−n) dim(δU) + 1

◮ If dim(X) exists, it is a countable ordinal and we call X countably dimensional. ◮ Otherwise X is infinite-dimensional.

A question from DST

Definition

Call Polish spaces X, Y piecewise homeomorphic, if there are partitions X =

i∈N Xi and Y = i∈N Yi such that

∀i ∈ N Xi ∼ = Yi. (denote this by X ∼ =ω Y)

Theorem (Hurewicz & Wallmann)

We have X ∼ =ω {0, 1}N iff X is countably dimensional.

Question (Jayne & Rogers; Motto-Ros, Schlicht & Selivanov)

Is there more than one ∼ =ω-equivalence class of infinite-dimensional Polish spaces?

Represented spaces and computability

Definition

A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.

Definition

F :⊆ NN → NN is a realizer of f : X → Y, iff δY(F(p)) = f(δX(p)) for all p ∈ δ−1

X (dom(f)). Abbreviate: F ⊢ f.

NN

F

− − − − → NN   δX   δY X

f

− − − − → Y

Definition

f : X → Y is called computable (continuous), iff it has a computable (continuous) realizer.

Computable metric spaces

We define a computable metric space with its Cauchy representation as follows:

Definition

• 1. An effective metric space is a tuple M = (M, d, (an)n∈N)

such that (M, d) is a metric space and (an)n∈N is a dense sequence in (M, d).

• 2. The Cauchy representation δM :⊆ NNM associated with the

effective metric space M = (M, d, (an)n∈N) is defined by δM(p) = x : ⇐ ⇒

• d(ap(i), ap(k)) ≤ 2−i for i < k

and x = lim

i→∞ ap(i)

Computable metric spaces, cont.

Theorem (Weihrauch)

For maps between effective metric spaces, metric continuity and realizer continuity coincide.

Question (Pour-El & Richards)

For which effective metric spaces do points have Turing degrees?

Point degree spectra

Definition

For A, B ⊆ NN, say A ≤M B iff ∃F : B → A, F computable.

Definition

For a represented space X, let Spec(X) = {δ−1

X (x)/ ≡M| x ∈ X} ⊆ M

be the point degree spectrum of X. ◮ Spec(NN) = Spec(R) = T (Turing degrees) ◮ Spec([0, 1]N) =: C (continuous degrees, MILLER) ◮ Spec(O(N)) = E (enumeration degrees)

Alternatively definition of the spectrum

Definition

For x ∈ X, y ∈ Y write x ≤T y iff there is a computable partial function F :⊆ Y → X with F(y) = x.

Definition

For x ∈ X, let ◮ Spec(x) = {p ∈ {0, 1}N | x ≤T p}. ◮ Spec(X) = {Spec(x) | x ∈ X} ◮ coSpec(x) = {p ∈ {0, 1}N | p ≤T x}. ◮ coSpec(X) = {coSpec(x) | x ∈ X} (works equivalently for countably based spaces)

What are the continuous degrees?

For A ∈ A({0, 1}N), let T(A) ⊆ {0, 1}N be all codes of trees for A.

Theorem

The following are equivalent for B ⊆ {0, 1}N:

• 1. ∃A ∈ A({0, 1}N)

B ≡M A ≡M T(A)

• 2. ∃X effective Polish, x ∈ X Spec(x) ≡M B

What are the continuous degrees?, contd

Theorem (MILLER)

If X is effective Polish and x ∈ X, then coSpec(x) is either a principal or a Scott-ideal.

Corollary

Let A ∈ A({0, 1}N) be such that A ≡M T(A) and A / ∈ T. If T(B) ≤M {p} ≤M A for B ∈ A({0, 1}N), p ∈ {0, 1}N, then B ≤M A.

Question

Is there a direct proof?

The main theorem

Theorem

The following are equivalent for a represented space X:

• 1. Spec(X) ⊆ Spec(Y)
• 2. X =

n∈N Xn where there are Yn ⊆ Y with Xn ∼

=c Yn

Theorem (Alternate form)

The following are equivalent for a represented space X:

• 1. ∃t ∈ T

t × Spec(X) = t × Spec(Y)

• 2. N × X ∼

=ω N × Y

Corollary

For Polish X the following are equivalent:

• 1. ∃p ∈ T p × Spec(X) ⊆ T
• 2. X has countable dimension.

A question and an answer

Question (MOTTO-ROS, SCHLICHT, SELIVANOV)

Is there a Polish space X s.t. for any p ∈ T we find that: T (p × Spec(X)) p × C

Theorem

There is an embedding of the inclusion ordering ([ω1]≤ω, ⊆) of countable subsets of the smallest uncountable ordinal ω1 into the piecewise-embeddability ordering of Polish spaces.

Towards the proof

Definition

An ω-left-CEA operator Jω

Ψ : {0, 1}N → [0, 1]N is generated by a

computable function Ψ : {0, 1}N × [0, 1]<ω × N2 → Q≥0 such that rn = lim sup

s→∞ Ψ(x, r0, . . . , rn−1, n, s)

Observation

There is an effective enumeration (Jω

e )e∈N of the ω-left-CEA

• perators.

Definition

The ω-left-computably-enumerable-in-and-above space ωCEA is a subspace of N × {0, 1}N × [0, 1]N defined by ωCEA = {(e, x, r) ∈ N × {0, 1}N × [0, 1]N : r = Jω

e (x)}

= “the graph of a universal ω-left-CEA operator.”

The first partial result

Theorem

• 1. ωCEA is a Polish space.
• 2. {0, 1}N ≇ω ωCEA ≇ω [0, 1]N

Side note:

Theorem

The following spaces are all piecewise homeomorphic to each

• ther.
• 1. The ω-left-CEA space ωCEA.
• 2. Rubin-Schori-Walsh’s totally disconnected strongly infinite

dimensional space RSW.

• 3. Roman Pol’s counterexample RP to Alexandrov’s problem.

The intuition

Figure: The upper and lower approximations of {0, 1}N, ωCEA and [0, 1]N

Some weird spaces

Theorem

There exists a nondegenerated continuum A ⊆ [0, 1]N in which no point has Turing degree.

Theorem (Following HUREWICZ)

The following are equivalent:

• 1. CH
• 2. There is an infinite dimensional space X ⊂ [0, 1]N, such

that any countably dimensional Z ⊂ X is countable.

• 3. There is an ascending chain Ω in the Turing degrees, such

that ∀t ∈ T ∃s ∈ Ω t ≤T s.

The lower reals

Definition

In R<, real numbers are represented as limits of increasing sequences of rationals.

Definition

U ∈ O(N) is called semi-recursive, if there is a computable function f : N × N → {0, 1} such that if n0 ∈ U ∨ n1 ∈ U, then nf(n0,n1) ∈ U. Let S ⊆ E be all degrees of semi-recursive sets.

Proposition (GANCHEV & SOSKOVA)

Spec(R<) = S

An application

Theorem

Let X be a countably-based T1 space. Then Rn+1

<

does not piecewise embed into X × Rn

<.

Let Λn = ({0, 1}n, ≤) be a partial order on {0, 1}n obtained as the n-th product of the ordering 0 < 1.

Lemma

For every countable partition (Pi)i∈ω of the n-dimensional hypercube [0, 1]n (endowed with the standard product order), there is i ∈ ω such that Pi has a subset which is order isomorphic to the product order Λn.

Corollary

For any n ∈ N there exists an enumeration degree which is expressible as the product of n + 1 semirecursive degrees, but not of n semirecursive degrees.

Outlook

◮ What other degree structures arise? (Partial answer: Looking at non-countably based spaces gets us beyond enumeration degrees) ◮ Which recursion-theoretic results hold there? (Kihara & Ng: The Shore-Slaman join theorem holds in the degree spectrum of any Polish space) ◮ Are there connections to other topological properties?

The article

• T. Kihara & A. Pauly.

Point degree spectra of represented spaces. arXiv 1405.6866, 2014.